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ABSTRACT. We propose in this paper an optimum turbo channel estimation algorithm for OFDM systems on highly se- lective fading channels. This algorithm ...
TURBO CHANNEL ESTIMATION FOR OFDM SYSTEMS ON HIGHLY TIME AND FREQUENCY SELECTIVE CHANNELS Emmanuel JAFFROT,Mohamed SIALA 38-40 Rue du General Leclerc, 92794 Issy Moulineaux Cedex [email protected], [email protected] ABSTRACT We propose in this paper an optimum turbo channel estimation algorithm for OFDM systems on highly selective fading channels. This algorithm performs an iterative estimation of the multiplicative fading channel according t o the maximum a posteriori criterion. It requires a convenient representation of the highly time and frequency selective fading channel based on the Karhunen-Loeve expansion theorem. It also optimally takes into account the coded structure of data symbols in the estimation process thanks t o the Bahl algorithm. 1. INTRODUCTION

We propose in this paper an optimum block-by-block two-dimensional turbo channel estimation algorithm for OFDM systems. This algorithm performs an iterative channel estimation according to the maximum a posteriori (MAP) criterion, using the Expecta-tion-Maximization (EM) algorithm [I]. It uses profitably both pilot (or reference) symbols and information-carrying symbols in the optimization of channel estimation [2]. In addition, it profits from the coded structure of the transmitted information-carrying symbols for an additional improvement of system performance. It requires a convenient representation of the multiplicative frequency and time selective fading channel using a Karhunen-LoGve (KL) expansion [3Jof the time-varying transfer function of the channel. This expansion relies on the spaced-frequency spaced-time correlation function of the two-dimensional fading channel [3]. The evaluation of the performance of this algorithm is based on an OFDM system with reception diversity.

with energy E,, and two-dimensional frequency-time position (mF,n T ) ,where F and T are respectively the frequency and time spacing between two adjacent symbols. These symbols take their values in an arbitrary PSK alphabet set R and are composed of NO data symbols with (two-dimensional) indices in a set S, and N p pilot symbols with indices in a set Sp. 3. MULTIPLICATIVE TWO-DIMENSIONAL FADING CHANNEL CHARACTERISTICS

We consider a rnulticarrier system with L decorrelated diversity branches. The time-variant transfer function [3] of the frequency and time selective fading channel seen at a given branch is characterized by its spacedfrequency spaced-time correlation function (SFSTCF)

4 (U,W. For a channel scattering function [3] with classical Doppler power spectrum (DPS) and exponential multipath intensity profile, the SFSTCF, with average power 4 (0,0 ) , is given by

where B d and T , are respectively the Doppler and multipath spreads of the channel and JO (.) is the Oth-order Bessel function of the first kind. 4. SIGNAL MODEL AT THE OUTPUT OF

THE RECEIVER MATCHED FILTER As depicted in Figure 1, the multicarrier receiver is composed of L diversity branches provided by spatially decorrelated receiving antennas. We assume that the lth diversity branch output signal associated to the symbol umn can be written as

2. TRANSMITTED SIGNAL

CHARACTERISTICS

1

Rmn = Cl,namn

We consider a block-by-block two-dimensional channel estimation for an OFDM system using PSK-modulated symbols. Each block is composed of N symbols umn

0-7803 - 6293-4/00/$10.00 02000 IEEE.

+ Nhn,

where ckn is the discrete channel gain factor of the l t h branch seen by the symbol amn and NL, is a complex

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where

Bp6(k)

is the kth component of B, and

can be computed by

We use where

as pth component of the initial guess GI('), where D q k ) is the value taken by the pilot symbol A , j ( k ) , b ( k ) E Sp.

P ( m1")

=P

(sk

= m I s k - 1 = m' )

are the known transitions probabilities of the code trellis and

7. CONDITIONAL PROBABILITIES COMPUTATION

q ( A Im', m ) = P

(Ak

= A 1 S k - i = m', s

k

=m)

are the coded symbols output probabilities. The normalized transmitted vector A is assumed to be For each iteration of the channel estimation algocoded using a trellis modulation. Therefore, the condirithm, the Bahl algorithm computes the aforementioned = A { Rl};;' , { G l ( d ) } L i l )probabilities using tional probabilities P can be computed by means of the Bahl algorithm [4]. This algorithm introduces the €unctions :

I

and

of the trellis at time k and m = 0 , . . . ,M - 1 the M distinct states of the code trellis. These functions can be computed respectively by the forward recursion.

8. SIMULATION RESULTS

s k denoting the state

We present below some simulation results for a 16carrier OFDM system with symbols provided by a trellis coded modulation whose trellis description is given in Figure 2.

M-1

ak ( m )=

ak-1

(m')?'k (m',m ) >

m'=O

with initial values

cro (0) = 1 and

00

( m )= 0 for m # 0,

and the backward recursion M-1 Pk

( m )=

Yk+l

(m',m)P k + l

("1

118

>

m'=O

Figure 2: Trellis coded modulation used in the simulations.

with inital values pjvD+l (0)

= 1 and

The function '?'k

(m',m ) =

P,vD+l

(m)= 0 for m # 0.

We restrict our investigation to time-frequency blocks with 256 symbols including 16 pilot symbols. The positions of these pilot symbols within each time-frequency block is specified in Figure 3. Moreover, we assume that all data and pilot symbols have a common energy E.

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Pilot symbols

1oo

n Data Symbols 3

...........

0

t

k

............... LL

m w

.............. ...............

loJ

Time

Figure 3: Pilot symbols positions in a time-frequency block. For illustration, we evaluate the performance of our algorithm for 2 severe channels with BdT, = lop3 and and L = 4 diversity branches. We use a channel scattering function with classical DPS and exponential multipath intensity profile and use a flat model for this function at the receiver. For the sake of simplicity, we restrict the number of iterations D carried out by our algorithm to 5. This number is deemed to be sufficient for almost reaching the best asymptotic achievable Bit Error Rate (BER). The performance of our algorithm is compared t o the performance of the hypothetical receiver with perfect channel knowledge. 1oo

................................................

:. ..................

.............................. lo-2/

-.

1o4

1: : 0

..I

j

..

.....*......

............................

.,

2

4

SNR

10

12

14

9. CONCLUSION We have proposed an iterative receiver with spatial diversity using an optimum turbo estimation of the multipath fading channel. We have noticed BER curves saturation for large values of the BdT, product and an increased degradation in performance with respect to the perfect channel estimation at low BER. A step forward t o reducing these phenomena will be by introducing time-frequency interleaving of coded data symbols and an enhanced initialisation of the estimation algorithm.

I

................. .......................

!:: : :: ::2:: ::: : : :::i . ::: : ::::I

potentially be gained by interleaving.

i ....I

. : . . . . . .:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . ..

. . . . .:. . . . . . . . . .:. ......... :. . . . . . . . . f .

Figure 5: Proposed channel estimator BER for BdT, = 10-~ and L = 4.

. . ...................... ......,........ . . .. . . . . . . . . . . .... .. .. .. . . . . . . ............ . . . ... ...... .........................

LL

ol:+

.

I

lod'

1

REFERENCES

A. P. DEMPSTER,N. M. LAIRD,and D. B. RUBIN, Maximum Likelihood from Incomplete Data via the EM algorithm, Journal of the Royal Statistical Society, Ser. 39, 1977.

} ...........................................

io"'

0

' . . . . . . . ....................

...{

M. SIALA,E. JAFFROT, Semi-Blind Maxamvm a Posteriori Fast Fading Channel Estimation for Multicarrier Systems, GRETSI 99, September 99, Vannes, France.

J 2

4

SNR

10

12

14

Figure 4: Proposed channel estimat,or BER for BdT, = and L = 4.

J. G. PROAKIS,Digital Communications, McGraw Hill, New York, 1989.

The simulations results show saturation for bot,h cases. This behaviour can be explained by the sensitivity of this algorithm t o initial guess quality. For BdT, = lop3,the saturation is even more pronounced due t o the lack of time and frequency diversity that can

L. R. BAHL,J. COCKE,F . JELINEK and J. RAVIV, Optimal Decoding of Linear Codes for Minimizing Symbol Error Rate, IEEE Transactions on Information Theory, vol. IT-20, March 1974.

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